There's one quantity that's going to come out the same,

但是也有一个量是始终不变的
no matter who is looking at the vector.

不管是谁在观察这个矢量
Anybody have a guess? Yes?

有人可以猜一下吗你说
The length of the vector is the length of the vector.

这个矢量的模长是保持不变的
Perhaps it's clear to you

这应该是比较显然的
that no matter how your axes are oriented,

无论坐标轴指向哪里
when you ask, "How long is this arrow?"

当你问 "这个矢量有多长"
you are going to get the same answer.

你都会得到相同的结果
That means (Ax)^2 + (Ay)^2 = (Ax')^2 + (Ay')^2.

也就是 (Ax)^2 + (Ay)^2 = (Ax')^2 + (Ay')^2
That's a property of this expression.

这是这个式子的一个性质
You can square this guy and add it to the square of this guy and

你可以把这两个式子平方相加
you will find, using the magic of trigonometry,

再用一下三角函数的性质